Explaining it really isn't that difficult. Let's say we have 60 people in a room. Random distribution says that our average is 5 birthdays each month.
In any given month, let's use 30 days for the day of the month. Our first birthday can be on any day. The second birthday has a 1 in 30 chance of being on that same day. The 3rd birthday has a 1 in 30 chance of being on the same day. The 4th birthday has a 1 in 30 chance of being on the same day. The 5th birthday has a 1 in 30 chance of being on the same day.
There's a 4 in 30 chance that somebody has the same birthday as the first guy.
With 5 guys, there's still the chance that the other 4 have birthdays on the same day, even if they don't share them with the first guy. I'm thinking the chance would be 4/30 + 3/30 + 2/30 + 1/30 for a total 10/30, or 1 in 3 chance of two people sharing the same birthday, in the first month...
With 12 months, you have 12 * 1 in 3 chances that out of a group of 60, some two people share birthdays -- and if that's not 100%, it's close enough to it to call it that! ;)